let X be bounded BCK-algebra; :: thesis: ( X is involutory iff for a being Element of X st a is being_greatest holds
for x, y being Element of X holds x \ (a \ y) = y \ (a \ x) )
thus ( X is involutory implies for a being Element of X st a is being_greatest holds
for x, y being Element of X holds x \ (a \ y) = y \ (a \ x) ) :: thesis: ( ( for a being Element of X st a is being_greatest holds
for x, y being Element of X holds x \ (a \ y) = y \ (a \ x) ) implies X is involutory )

proof

assume A1: X is involutory ; :: thesis: for a being Element of X st a is being_greatest holds
for x, y being Element of X holds x \ (a \ y) = y \ (a \ x)
let a be Element of X; :: thesis: ( a is being_greatest implies for x, y being Element of X holds x \ (a \ y) = y \ (a \ x) )
assume A2: a is being_greatest ; :: thesis: for x, y being Element of X holds x \ (a \ y) = y \ (a \ x)
let x, y be Element of X; :: thesis: x \ (a \ y) = y \ (a \ x)
( x \ (a \ y) = (a \ (a \ y)) \ (a \ x) & y \ (a \ x) = (a \ (a \ x)) \ (a \ y) ) by A1, A2, Th22;
hence x \ (a \ y) = y \ (a \ x) by BCIALG_1:7; :: thesis: verum

end;

assume A3: for a being Element of X st a is being_greatest holds
for x, y being Element of X holds x \ (a \ y) = y \ (a \ x) ; :: thesis: X is involutory
let a be Element of X; :: according to BCIALG_3:def 7 :: thesis: ( a is being_greatest implies for x being Element of X holds a \ (a \ x) = x )
assume A4: a is being_greatest ; :: thesis: for x being Element of X holds a \ (a \ x) = x
let x be Element of X; :: thesis: a \ (a \ x) = x
a \ (a \ x) = x \ (a \ a) by A3, A4
.= x \ (0. X) by BCIALG_1:def 5
.= x by BCIALG_1:2 ;
hence a \ (a \ x) = x ; :: thesis: verum